WEBVTT 00:00:07.930 --> 00:00:13.530 OK, here we go with, quiz review for the third quiz that's 00:00:13.530 --> 00:00:15.910 coming on Friday. 00:00:15.910 --> 00:00:19.040 So, one key point is that the quiz 00:00:19.040 --> 00:00:25.120 covers through chapter six. 00:00:25.120 --> 00:00:27.580 Chapter seven on linear transformations 00:00:27.580 --> 00:00:31.930 will appear on the final exam, but not on the quiz. 00:00:31.930 --> 00:00:35.540 So I won't review linear transformations today, 00:00:35.540 --> 00:00:39.720 but they'll come into the full course review on the very 00:00:39.720 --> 00:00:41.350 last lecture. 00:00:41.350 --> 00:00:44.120 So today, I'm reviewing chapter six, 00:00:44.120 --> 00:00:46.330 and I'm going to take some old exams, 00:00:46.330 --> 00:00:48.990 and I'm always ready to answer questions. 00:00:48.990 --> 00:00:54.360 And I thought, kind of help our memories if I write down 00:00:54.360 --> 00:01:00.350 the main topics in chapter six. 00:01:00.350 --> 00:01:04.019 So, already, on the previous quiz, 00:01:04.019 --> 00:01:07.520 we knew how to find eigenvalues and eigenvectors. 00:01:07.520 --> 00:01:13.480 Well, we knew how to find them by that determinant of A minus 00:01:13.480 --> 00:01:15.390 lambda I equals zero. 00:01:15.390 --> 00:01:17.280 But, of course, there could be shortcuts. 00:01:17.280 --> 00:01:20.030 There could be, like, useful information 00:01:20.030 --> 00:01:27.190 about the eigenvalues that we can speed things up with. 00:01:27.190 --> 00:01:27.830 OK. 00:01:27.830 --> 00:01:32.300 Then, the new stuff starts out with a differential equation, 00:01:32.300 --> 00:01:34.680 so I'll do a problem. 00:01:34.680 --> 00:01:36.810 I'll do a differential equation problem first. 00:01:39.970 --> 00:01:43.010 What's special about symmetric matrices? 00:01:43.010 --> 00:01:46.060 Can we just say that in words? 00:01:46.060 --> 00:01:48.170 I'd better write it down, though. 00:01:48.170 --> 00:01:50.960 What's special about symmetric matrices? 00:01:50.960 --> 00:01:56.970 Their eigenvalues are real. 00:01:56.970 --> 00:02:00.970 The eigenvalues of a symmetric matrix always come out real, 00:02:00.970 --> 00:02:04.800 and there always are enough eigenvectors. 00:02:04.800 --> 00:02:06.930 Even if there are repeated eigenvalues, 00:02:06.930 --> 00:02:09.910 there are enough eigenvectors, and we 00:02:09.910 --> 00:02:13.180 can choose those eigenvectors to be orthogonal. 00:02:13.180 --> 00:02:16.480 So if A equals A transposed, the big fact 00:02:16.480 --> 00:02:22.620 will be that we can diagonalize it, 00:02:22.620 --> 00:02:28.550 and those eigenvector matrix, with the eigenvectors 00:02:28.550 --> 00:02:31.470 in the column, can be an orthogonal matrix. 00:02:31.470 --> 00:02:36.600 So we get a Q lambda Q transpose. 00:02:36.600 --> 00:02:43.010 That, in three symbols, expresses a wonderful fact, 00:02:43.010 --> 00:02:46.250 a fundamental fact for symmetric matrices. 00:02:46.250 --> 00:02:46.960 OK. 00:02:46.960 --> 00:02:49.740 Then, we went beyond that fact to ask 00:02:49.740 --> 00:02:52.570 about positive definite matrices, when 00:02:52.570 --> 00:02:54.370 the eigenvalues were positive. 00:02:54.370 --> 00:02:55.970 I'll do an example of that. 00:02:59.290 --> 00:03:01.580 Now we've left symmetry. 00:03:01.580 --> 00:03:05.000 Similar matrices are any square matrices, 00:03:05.000 --> 00:03:10.730 but two matrices are similar if they're related that way. 00:03:10.730 --> 00:03:14.590 And what's the key point about similar matrices? 00:03:14.590 --> 00:03:16.770 Somehow, those matrices are representing 00:03:16.770 --> 00:03:19.410 the same thing in different basis, 00:03:19.410 --> 00:03:21.980 in chapter seven language. 00:03:21.980 --> 00:03:29.220 In chapter six language, what's up with these similar matrices? 00:03:29.220 --> 00:03:31.860 What's the key fact, the key positive fact 00:03:31.860 --> 00:03:33.570 about similar matrices? 00:03:33.570 --> 00:03:37.130 They have the same eigenvalues. 00:03:37.130 --> 00:03:39.070 Same eigenvalues. 00:03:39.070 --> 00:03:42.460 So if one of them grows, the other one grows. 00:03:42.460 --> 00:03:50.380 If one of them decays to zero, the other one decays to zero. 00:03:50.380 --> 00:03:53.050 Powers of A will look like powers of B, 00:03:53.050 --> 00:03:55.440 because powers of A and powers of B 00:03:55.440 --> 00:03:59.160 only differ by an M inverse and an M way on the 00:03:59.160 --> 00:04:00.040 outside. 00:04:00.040 --> 00:04:05.250 So if these are similar, then B to the k-th power 00:04:05.250 --> 00:04:09.160 is M inverse A to the k-th power M. 00:04:09.160 --> 00:04:12.410 And that's why I say, eh, this M, it 00:04:12.410 --> 00:04:14.480 does change the eigenvectors, but it 00:04:14.480 --> 00:04:16.089 doesn't change the eigenvalues. 00:04:16.089 --> 00:04:21.209 So same lambdas. 00:04:21.209 --> 00:04:24.510 And then, finally, I've got to review the point 00:04:24.510 --> 00:04:30.000 about the SVD, the Singular Value Decomposition. 00:04:30.000 --> 00:04:30.500 OK. 00:04:30.500 --> 00:04:33.900 So that's what this quiz has got to cover, 00:04:33.900 --> 00:04:37.720 and now I'll just take problems from earlier exams, 00:04:37.720 --> 00:04:40.460 starting with a differential equation. 00:04:40.460 --> 00:04:41.250 OK. 00:04:41.250 --> 00:04:43.360 And always ready for questions. 00:04:43.360 --> 00:04:46.640 So here is an exam from about the year 00:04:46.640 --> 00:04:53.290 zero, and it has a three by three. 00:04:53.290 --> 00:04:54.502 So that was -- 00:04:57.280 --> 00:04:59.720 but it's a pretty special-looking matrix, 00:04:59.720 --> 00:05:04.190 it's got zeroes on the diagonal, it's got minus ones above, 00:05:04.190 --> 00:05:08.360 and it's got plus ones like that. 00:05:08.360 --> 00:05:12.160 So that's the matrix A. 00:05:12.160 --> 00:05:12.880 OK. 00:05:12.880 --> 00:05:16.750 Step one is, well, I want to solve that 00:05:16.750 --> 00:05:17.700 equation. 00:05:17.700 --> 00:05:19.970 I want to find the general solution. 00:05:19.970 --> 00:05:22.790 I haven't given you a u(0) here, so I'm 00:05:22.790 --> 00:05:24.900 looking for the general solution, 00:05:24.900 --> 00:05:28.280 so now what's the form of the general solution? 00:05:28.280 --> 00:05:30.520 With three arbitrary constants going 00:05:30.520 --> 00:05:33.940 to be inside it, because those will be used 00:05:33.940 --> 00:05:35.780 to match the initial condition. 00:05:35.780 --> 00:05:39.580 So the general form is u at time t 00:05:39.580 --> 00:05:45.500 is some multiple of the first special solution. 00:05:45.500 --> 00:05:50.080 The first special solution will be growing like the eigenvalue, 00:05:50.080 --> 00:05:51.920 and it's the eigenvector. 00:05:51.920 --> 00:05:56.400 So that's a pure exponential solution, just staying 00:05:56.400 --> 00:05:58.580 with that eigenvector. 00:05:58.580 --> 00:06:01.770 Of course, I haven't found, yet, the eigenvalues 00:06:01.770 --> 00:06:02.580 and eigenvectors. 00:06:02.580 --> 00:06:04.900 That's, normally, the first job. 00:06:04.900 --> 00:06:08.880 Now, there will be second one, growing like e 00:06:08.880 --> 00:06:13.160 to the lambda two, and a third one growing like e 00:06:13.160 --> 00:06:16.450 to the lambda three. 00:06:16.450 --> 00:06:18.880 So we're all done -- 00:06:18.880 --> 00:06:20.860 well, we haven't done anything yet, 00:06:20.860 --> 00:06:22.100 actually. 00:06:22.100 --> 00:06:25.930 I've got to find the eigenvalues and eigenvectors, 00:06:25.930 --> 00:06:30.720 and then I would match u(0) by choosing the right three 00:06:30.720 --> 00:06:31.420 constants. 00:06:31.420 --> 00:06:31.950 OK. 00:06:31.950 --> 00:06:35.030 So now I ask -- ask you about the eigenvalues 00:06:35.030 --> 00:06:38.790 and eigenvectors, and you look at this matrix and what do you 00:06:38.790 --> 00:06:41.390 see in that matrix? 00:06:41.390 --> 00:06:47.740 Um, well, I guess we might ask ourselves right away, 00:06:47.740 --> 00:06:50.760 is it singular? 00:06:50.760 --> 00:06:51.810 Is it singular? 00:06:51.810 --> 00:06:54.040 Because, if so, then we really have a head start, 00:06:54.040 --> 00:06:56.810 we know one of the eigenvalues is zero. 00:06:56.810 --> 00:06:57.990 Is that matrix singular? 00:07:01.660 --> 00:07:04.280 Eh, I don't know, do you take the determinant to 00:07:04.280 --> 00:07:05.110 find out? 00:07:05.110 --> 00:07:08.530 Or maybe you look at the first row and third row 00:07:08.530 --> 00:07:10.270 and say, hey, the first row and third row 00:07:10.270 --> 00:07:15.180 are just opposite signs, they're linear-dependent? 00:07:15.180 --> 00:07:17.930 The first column and third column are dependent -- it's 00:07:17.930 --> 00:07:19.150 singular. 00:07:19.150 --> 00:07:22.610 So one eigenvalue is zero. 00:07:22.610 --> 00:07:24.520 Let's make that lambda one. 00:07:24.520 --> 00:07:26.150 Lambda one, then, will be zero. 00:07:26.150 --> 00:07:26.850 OK. 00:07:26.850 --> 00:07:30.000 Now we've got a couple of other eigenvalues to find, 00:07:30.000 --> 00:07:33.430 and, I suppose the simplest way is 00:07:33.430 --> 00:07:36.940 to look at A minus lambda I So let 00:07:36.940 --> 00:07:43.720 me just put minus lambda in here, minus ones above, 00:07:43.720 --> 00:07:45.520 ones below. 00:07:45.520 --> 00:07:51.430 But, actually, before I do it, that matrix 00:07:51.430 --> 00:07:53.910 is not symmetric, for sure, right? 00:07:53.910 --> 00:07:57.220 In fact, it's the very opposite of symmetric. 00:07:57.220 --> 00:08:02.420 That matrix A transpose, how is A transpose connected to A? 00:08:02.420 --> 00:08:04.020 It's negative A. 00:08:04.020 --> 00:08:07.890 It's an anti-symmetric matrix, skew-symmetric matrix. 00:08:07.890 --> 00:08:11.230 And we've met, maybe, a two-by-two example 00:08:11.230 --> 00:08:14.370 of skew-symmetric matrices, and let 00:08:14.370 --> 00:08:17.687 me just say, what's the deal with their eigenvalues? 00:08:17.687 --> 00:08:18.645 They're pure imaginary. 00:08:21.400 --> 00:08:23.440 They'll be on the imaginary axis, 00:08:23.440 --> 00:08:26.710 there be some multiple of I if it's 00:08:26.710 --> 00:08:29.090 an anti-symmetric, skew-symmetric matrix. 00:08:29.090 --> 00:08:31.360 So I'm looking for multiples of I, 00:08:31.360 --> 00:08:33.730 and of course, that's zero times I, 00:08:33.730 --> 00:08:37.760 that's on the imaginary axis, but maybe I just do it out, 00:08:37.760 --> 00:08:38.350 here. 00:08:38.350 --> 00:08:39.740 Lambda cubed. 00:08:39.740 --> 00:08:42.600 well, maybe that's minus lambda cubed, 00:08:42.600 --> 00:08:44.480 and then a zero and a zero. 00:08:44.480 --> 00:08:47.870 Zero, and then maybe I have a plus a lambda, 00:08:47.870 --> 00:08:52.610 and another plus lambda, but those go with a minus sign. 00:08:52.610 --> 00:08:55.080 Am I getting minus two lambda equals zero? 00:08:58.640 --> 00:08:59.940 So. 00:08:59.940 --> 00:09:04.810 So I'm solving lambda cube plus two lambda equals zero. 00:09:04.810 --> 00:09:09.070 So one root factors out lambda, and the the rest 00:09:09.070 --> 00:09:11.571 is lambda squared plus two. 00:09:11.571 --> 00:09:12.070 OK. 00:09:12.070 --> 00:09:14.310 This is going the way we expect, right? 00:09:14.310 --> 00:09:22.360 Because this gives the root lambda equals zero, and gives 00:09:22.360 --> 00:09:27.630 the other two roots, which are lambda equal what? 00:09:27.630 --> 00:09:30.200 The solutions of when is lambda squared 00:09:30.200 --> 00:09:36.510 plus two equals zero then the eigenvalues those guys, what 00:09:36.510 --> 00:09:38.610 are they? 00:09:38.610 --> 00:09:41.260 They're a multiple of i, they're just square root of two 00:09:41.260 --> 00:09:42.150 i. 00:09:42.150 --> 00:09:45.010 When I set this equals to zero, I 00:09:45.010 --> 00:09:48.800 have lambda squared equal to minus two, right? 00:09:48.800 --> 00:09:50.710 To make that zero? 00:09:50.710 --> 00:09:54.180 And the roots are square root of two i 00:09:54.180 --> 00:09:58.060 and minus the square root of two i. 00:09:58.060 --> 00:09:59.540 So now I know what those are. 00:09:59.540 --> 00:10:01.070 I'll put those in, now. 00:10:01.070 --> 00:10:03.840 Either the zero t is just a one. 00:10:03.840 --> 00:10:06.840 That's just a one. 00:10:06.840 --> 00:10:13.700 This is square root of two I and this is minus 00:10:13.700 --> 00:10:18.280 square root of two I. 00:10:18.280 --> 00:10:22.590 So, is the solution decaying to zero? 00:10:22.590 --> 00:10:25.210 Is this a completely stable problem 00:10:25.210 --> 00:10:28.630 where the solution is going to zero? 00:10:28.630 --> 00:10:31.200 No. 00:10:31.200 --> 00:10:34.620 In fact, all these things are staying the same size. 00:10:34.620 --> 00:10:39.290 This thing is getting multiplied by this number. 00:10:39.290 --> 00:10:46.880 e to the I something t, that's a number that has magnitude one, 00:10:46.880 --> 00:10:50.160 and sort of wanders around the unit circle. 00:10:50.160 --> 00:10:51.660 Same for this. 00:10:51.660 --> 00:10:55.900 So that the solution doesn't blow up, and it doesn't go to 00:10:55.900 --> 00:10:56.520 zero. 00:10:56.520 --> 00:10:57.410 OK. 00:10:57.410 --> 00:10:59.670 And to find out what it actually is, 00:10:59.670 --> 00:11:02.370 we would have to plug in initial conditions. 00:11:02.370 --> 00:11:04.410 But actually, the next question I ask 00:11:04.410 --> 00:11:12.280 is, when does the solution return to its initial value? 00:11:12.280 --> 00:11:15.910 I won't even say what's the initial value. 00:11:15.910 --> 00:11:22.540 This is a case in which I think this solution is 00:11:22.540 --> 00:11:25.350 periodic after. 00:11:25.350 --> 00:11:31.820 At t equals zero, it starts with c1, c2, and c3, 00:11:31.820 --> 00:11:36.500 and then at some value of t, it comes back to that. 00:11:36.500 --> 00:11:38.440 So that's a very special question, 00:11:38.440 --> 00:11:40.280 Well, let's just take three seconds, 00:11:40.280 --> 00:11:44.040 because that special question isn't likely to be on the quiz. 00:11:44.040 --> 00:11:49.910 But it comes back to the start, when? 00:11:49.910 --> 00:11:55.230 Well, whenever we have e to the two pi i, that's one, 00:11:55.230 --> 00:11:56.450 and we've come back again. 00:11:56.450 --> 00:11:58.700 So it comes back to the start. 00:11:58.700 --> 00:12:07.740 It's periodic, when this square root of two i -- 00:12:07.740 --> 00:12:10.930 shall I call it capital T, for the period? 00:12:10.930 --> 00:12:17.270 For that particular T, if that equals two pi i, then e 00:12:17.270 --> 00:12:21.090 to this thing is one, and we've come around again. 00:12:21.090 --> 00:12:26.250 So the period is T is determined here, cancel the i-s, 00:12:26.250 --> 00:12:30.970 and T is pi times the square root of two. 00:12:30.970 --> 00:12:32.270 So that's pretty neat. 00:12:32.270 --> 00:12:35.610 We get all the information about all solutions, 00:12:35.610 --> 00:12:39.230 we haven't fixed on only one particular solution, 00:12:39.230 --> 00:12:41.450 but it comes around again. 00:12:41.450 --> 00:12:43.730 So this was probably my first chance 00:12:43.730 --> 00:12:46.270 to say something about the whole family 00:12:46.270 --> 00:12:49.760 of anti-symmetric, skew-symmetric matrices. 00:12:49.760 --> 00:12:50.380 OK. 00:12:50.380 --> 00:12:56.880 And then, finally, I asked, take two eigenvectors (again, 00:12:56.880 --> 00:12:59.450 I haven't computed the eigenvectors) 00:12:59.450 --> 00:13:02.690 and it turns out they're orthogonal. 00:13:02.690 --> 00:13:03.500 They're orthogonal. 00:13:03.500 --> 00:13:06.500 The eigenvectors of a symmetric matrix, 00:13:06.500 --> 00:13:10.250 or a skew-symmetric matrix, are always orthogonal. 00:13:13.200 --> 00:13:18.020 I guess may conscience makes me tell you, 00:13:18.020 --> 00:13:23.950 what are all the matrices that have orthogonal eigenvectors? 00:13:23.950 --> 00:13:27.040 And symmetric is the most important class, 00:13:27.040 --> 00:13:28.640 so that's the one we've spoken about. 00:13:28.640 --> 00:13:32.800 But let me just put that little fact down, here. 00:13:32.800 --> 00:13:36.661 Orthogonal x-s. 00:13:36.661 --> 00:13:37.202 eigenvectors. 00:13:41.430 --> 00:13:44.200 A matrix has orthogonal eigenvectors, 00:13:44.200 --> 00:13:47.430 the exact condition -- it's quite beautiful that I can tell 00:13:47.430 --> 00:13:49.340 you exactly when that happens. 00:13:49.340 --> 00:13:55.290 It happens when A times A transpose equals A transpose 00:13:55.290 --> 00:14:00.290 times A. Any time that's the condition 00:14:00.290 --> 00:14:03.700 for orthogonal eigenvectors. 00:14:03.700 --> 00:14:09.040 And because we're interested in special families of vectors, 00:14:09.040 --> 00:14:12.690 tell me some special families that fit. 00:14:12.690 --> 00:14:15.410 This is the whole requirement. 00:14:15.410 --> 00:14:21.120 That's a pretty special requirement most matrices have. 00:14:21.120 --> 00:14:23.070 So the average three-by-three matrix 00:14:23.070 --> 00:14:26.240 has three eigenvectors, but not orthogonal. 00:14:26.240 --> 00:14:29.380 But if it happens to commute with its transpose, 00:14:29.380 --> 00:14:34.010 then, wonderfully, the eigenvectors are orthogonal. 00:14:34.010 --> 00:14:39.250 Now, do you see how symmetric matrices pass this test? 00:14:39.250 --> 00:14:40.340 Of course. 00:14:40.340 --> 00:14:43.790 If A transpose equals A, then both sides are A squared, 00:14:43.790 --> 00:14:45.640 we've got it. 00:14:45.640 --> 00:14:49.410 How do anti-symmetric matrices pass this test? 00:14:49.410 --> 00:14:54.060 If A transpose equals minus A, then we've 00:14:54.060 --> 00:14:58.220 got it again, because we've got minus A squared on both sides. 00:14:58.220 --> 00:15:00.050 So that's another group. 00:15:00.050 --> 00:15:03.090 And finally, let me ask you about our other favorite 00:15:03.090 --> 00:15:06.950 family, orthogonal matrices. 00:15:06.950 --> 00:15:11.730 Do orthogonal matrices pass this test, if A is a Q, 00:15:11.730 --> 00:15:14.990 do they pass the test for orthogonal eigenvectors. 00:15:14.990 --> 00:15:22.320 Well, if A is Q, an orthogonal matrix, what is Q transpose Q? 00:15:22.320 --> 00:15:23.020 It's I. 00:15:23.020 --> 00:15:25.310 And what is Q Q transpose? 00:15:25.310 --> 00:15:28.010 It's I, we're talking square matrices here. 00:15:28.010 --> 00:15:30.100 So yes, it passes the test. 00:15:30.100 --> 00:15:36.600 So the special cases are symmetric, anti-symmetric 00:15:36.600 --> 00:15:41.080 (I'll say skew-symmetric,) and orthogonal. 00:15:41.080 --> 00:15:44.280 Those are the three important special classes 00:15:44.280 --> 00:15:45.710 that are in this family. 00:15:45.710 --> 00:15:46.210 OK. 00:15:46.210 --> 00:15:52.860 That's like a comment that, could have been made back in, 00:15:52.860 --> 00:15:54.870 section six point four. 00:15:54.870 --> 00:16:04.390 OK, I can pursue the differential equations, also 00:16:04.390 --> 00:16:09.090 this question, didn't ask you to tell me, 00:16:09.090 --> 00:16:13.920 how would I find this matrix exponential, e to the At? 00:16:13.920 --> 00:16:15.050 So can I erase this? 00:16:15.050 --> 00:16:17.050 I'll just stay with this same... 00:16:19.690 --> 00:16:23.770 how would I find e to the At? 00:16:23.770 --> 00:16:27.010 Because, how does that come in? 00:16:27.010 --> 00:16:30.140 That's the key matrix for a differential equation, 00:16:30.140 --> 00:16:32.460 because the solution is -- 00:16:32.460 --> 00:16:38.520 the solution is u(t) is e^(At) u(0). 00:16:38.520 --> 00:16:42.060 So this is like the fundamental matrix 00:16:42.060 --> 00:16:48.400 that multiplies the given function and gives the answer. 00:16:48.400 --> 00:16:53.630 And how would we compute it if we wanted that? 00:16:53.630 --> 00:16:56.830 We don't always have to find e to the At, because I 00:16:56.830 --> 00:17:00.500 can go directly to the answer without any e to the At-s, 00:17:00.500 --> 00:17:06.440 but hiding here is an e to the At, and how would I compute it? 00:17:06.440 --> 00:17:10.026 Well, if A is diagonalizable. 00:17:12.589 --> 00:17:21.510 So I'm now going to put in my usual if A can be diagonalized 00:17:21.510 --> 00:17:25.240 (and everybody remember that there is an if there, 00:17:25.240 --> 00:17:28.820 because it might not have enough eigenvectors) 00:17:28.820 --> 00:17:33.330 this example does have enough, random matrices have enough. 00:17:33.330 --> 00:17:36.720 So if we can diagonalize, then we get a nice formula for this, 00:17:36.720 --> 00:17:40.050 because an S comes way out at the beginning, 00:17:40.050 --> 00:17:42.740 and S inverse comes way out at the end, 00:17:42.740 --> 00:17:47.450 and we only have to take the exponential of lambda. 00:17:47.450 --> 00:17:49.980 And that's just a diagonal matrix, 00:17:49.980 --> 00:17:53.780 so that's just e the lambda one t, 00:17:53.780 --> 00:18:00.030 these guys are showing up, now, in e to the lambda nt. 00:18:00.030 --> 00:18:01.070 OK? 00:18:01.070 --> 00:18:03.510 That's a really quick review of that formula. 00:18:06.040 --> 00:18:08.780 It's something we can compute it quickly 00:18:08.780 --> 00:18:11.250 if we have done the S and lambda part. 00:18:13.770 --> 00:18:15.470 If we know S and lambda, then it's 00:18:15.470 --> 00:18:17.440 not hard to take that step. 00:18:17.440 --> 00:18:20.770 OK, that's some comments on differential equations. 00:18:20.770 --> 00:18:28.330 I would like to go on to a next question that I started here. 00:18:28.330 --> 00:18:33.410 And it's, got several parts, and I can just read it out. 00:18:33.410 --> 00:18:37.320 What we're given is a three-by-three matrix, 00:18:37.320 --> 00:18:41.900 and we're told its eigenvalues, except one of these 00:18:41.900 --> 00:18:47.700 is, like, we don't know, and we're told the eigenvectors. 00:18:47.700 --> 00:18:50.500 And I want to ask you about the matrix. 00:18:50.500 --> 00:18:51.190 OK. 00:18:51.190 --> 00:18:54.530 So, first question. 00:18:54.530 --> 00:18:56.430 Is the matrix diagonalizable? 00:18:59.000 --> 00:19:03.030 And I really mean for which c, because I 00:19:03.030 --> 00:19:06.850 don't know c, so my questions will all be, 00:19:06.850 --> 00:19:12.340 for which is there a condition on c, does one c work. 00:19:12.340 --> 00:19:17.270 But your answer should tell me all the c-s that work. 00:19:17.270 --> 00:19:21.390 I'm not asking for you to tell me, well, c equal four, 00:19:21.390 --> 00:19:22.660 yes, that checks out. 00:19:22.660 --> 00:19:27.927 I want to know all the c-s that make it diagonalizable. 00:19:34.950 --> 00:19:36.640 OK? 00:19:36.640 --> 00:19:39.440 What's the real on diagonalizable? 00:19:39.440 --> 00:19:42.212 We need enough eigenvectors, right? 00:19:42.212 --> 00:19:43.920 We don't care what those eigenvalues are, 00:19:43.920 --> 00:19:46.710 it's eigenvectors that count for diagonalizable, 00:19:46.710 --> 00:19:49.220 and we need three independent ones, 00:19:49.220 --> 00:19:52.340 and are those three guys independent? 00:19:52.340 --> 00:19:53.580 Yes. 00:19:53.580 --> 00:19:56.300 Actually, let's look at them for a moment. 00:19:56.300 --> 00:19:59.921 What do you see about those three vectors right away? 00:19:59.921 --> 00:20:01.170 They're more than independent. 00:20:04.380 --> 00:20:09.730 Can you see why those three got chosen? 00:20:09.730 --> 00:20:15.780 Because it will come up in the next part, they're orthogonal. 00:20:15.780 --> 00:20:17.920 Those eigenvectors are orthogonal. 00:20:17.920 --> 00:20:19.710 They're certainly independent. 00:20:19.710 --> 00:20:28.220 So the answer to diagonalizable is, yes, all c, all c. 00:20:28.220 --> 00:20:30.760 Doesn't matter. c could be a repeated guy, 00:20:30.760 --> 00:20:32.360 but we've got enough eigenvectors, 00:20:32.360 --> 00:20:33.930 so that's what we care about. 00:20:33.930 --> 00:20:36.400 OK, second question. 00:20:36.400 --> 00:20:38.365 For which values of c is it symmetric? 00:20:40.960 --> 00:20:46.000 OK, what's the answer to that one? 00:20:48.650 --> 00:20:53.420 If we know the same setup if we know that much about it, 00:20:53.420 --> 00:20:55.330 we know those eigenvectors, and we've 00:20:55.330 --> 00:21:02.850 noticed they're orthogonal, then which c-s will work? 00:21:02.850 --> 00:21:07.800 So the eigenvalues of that symmetric matrix have to be 00:21:07.800 --> 00:21:08.490 real. 00:21:08.490 --> 00:21:11.780 So all real c. 00:21:11.780 --> 00:21:17.300 If c was i, the matrix wouldn't have been symmetric. 00:21:17.300 --> 00:21:24.040 But if c is a real number, then we've got real eigenvalues, 00:21:24.040 --> 00:21:25.920 we've got orthogonal eigenvectors, 00:21:25.920 --> 00:21:27.400 that matrix is symmetric. 00:21:27.400 --> 00:21:28.790 OK, positive definite. 00:21:28.790 --> 00:21:40.630 OK, now this is a sub-case of symmetric, 00:21:40.630 --> 00:21:45.900 so we need c to be real, so we've got a symmetric matrix, 00:21:45.900 --> 00:21:50.360 but we also want the thing to be positive definite. 00:21:50.360 --> 00:21:52.340 Now, we're looking at eigenvalues, 00:21:52.340 --> 00:21:54.740 we've got a lot of tests for positive definite, 00:21:54.740 --> 00:21:57.250 but eigenvalues, if we know them, 00:21:57.250 --> 00:22:01.100 is certainly a good, quick, clean test. 00:22:01.100 --> 00:22:05.570 Could this matrix be positive definite? 00:22:05.570 --> 00:22:06.640 No. 00:22:06.640 --> 00:22:10.180 No, because it's got an eigenvalue zero. 00:22:10.180 --> 00:22:12.640 It could be positive semi-definite, 00:22:12.640 --> 00:22:15.900 you know, like consolation prize, 00:22:15.900 --> 00:22:19.320 if c was greater or equal to zero, 00:22:19.320 --> 00:22:21.680 it would be positive semi-definite. 00:22:21.680 --> 00:22:25.520 But it's not, no. 00:22:25.520 --> 00:22:30.670 Semi-definite, if I put that comment in, semi-definite, 00:22:30.670 --> 00:22:35.010 that the condition would be c greater or equal to zero. 00:22:35.010 --> 00:22:36.300 That would be all right. 00:22:36.300 --> 00:22:37.220 OK. 00:22:37.220 --> 00:22:38.330 Next part. 00:22:38.330 --> 00:22:39.675 Is it a Markov matrix? 00:22:44.260 --> 00:22:44.870 Hm. 00:22:44.870 --> 00:22:50.440 Could this matrix be, if I choose the number c correctly, 00:22:50.440 --> 00:22:52.040 a Markov matrix? 00:22:58.750 --> 00:23:02.700 Well, what do we know about Markov matrices? 00:23:02.700 --> 00:23:05.320 Mainly, we know something about their eigenvalues. 00:23:05.320 --> 00:23:10.500 One eigenvalue is always one, and the other eigenvalues 00:23:10.500 --> 00:23:13.430 are smaller. 00:23:13.430 --> 00:23:14.690 Not larger. 00:23:14.690 --> 00:23:17.380 So an eigenvalue two can't happen. 00:23:17.380 --> 00:23:22.000 So the answer is, no, not a ma- that's never a Markov matrix. 00:23:22.000 --> 00:23:22.750 OK? 00:23:22.750 --> 00:23:29.970 And finally, could one half of A be a projection matrix? 00:23:29.970 --> 00:23:32.820 So could it- could this -- eh-eh could this be twice 00:23:32.820 --> 00:23:33.890 a projection matrix? 00:23:33.890 --> 00:23:35.840 So let me write it this way. 00:23:35.840 --> 00:23:39.530 Could A over two be a projection matrix? 00:23:44.820 --> 00:23:46.820 OK, what are projection matrices? 00:23:46.820 --> 00:23:48.650 They're real. 00:23:48.650 --> 00:23:53.160 I mean, th- they're symmetric, so their eigenvalues are real. 00:23:53.160 --> 00:23:56.680 But more than that, we know what those eigenvalues have to be. 00:23:56.680 --> 00:24:01.150 What do the eigenvalues of a projection matrix have to be? 00:24:01.150 --> 00:24:05.670 See, that any nice matrix we've got 00:24:05.670 --> 00:24:08.490 an idea about its eigenvalues. 00:24:08.490 --> 00:24:12.980 So the eigenvalues of projection matrices are zero and 00:24:12.980 --> 00:24:13.970 one. 00:24:13.970 --> 00:24:16.710 Zero and one, only. 00:24:16.710 --> 00:24:21.510 Because P squared equals P, let me call this matrix P, 00:24:21.510 --> 00:24:26.510 so P squared equals P, so lambda squared equals lambda, 00:24:26.510 --> 00:24:30.570 because eigenvalues of P squared are lambda squared, 00:24:30.570 --> 00:24:37.520 and we must have that, so lambda equals zero or one. 00:24:37.520 --> 00:24:38.140 OK. 00:24:38.140 --> 00:24:42.060 Now what value of c will work there? 00:24:42.060 --> 00:24:48.250 So, then, there are some value that will work, 00:24:48.250 --> 00:24:50.300 and what will work? 00:24:50.300 --> 00:24:56.340 c equals zero will work, or what else will work? 00:24:59.310 --> 00:25:02.690 c equal to two. 00:25:02.690 --> 00:25:06.260 Because if c is two, then when we divide by two, 00:25:06.260 --> 00:25:09.880 this Eigenvalue of two will drop to one, 00:25:09.880 --> 00:25:13.110 and so will the other one, so, or c equal to two. 00:25:13.110 --> 00:25:15.640 OK, those are the guys that will work, 00:25:15.640 --> 00:25:21.110 and it was the fact that those eigenvectors were orthogonal, 00:25:21.110 --> 00:25:23.780 the fact that those eigenvectors were orthogonal 00:25:23.780 --> 00:25:26.170 carried us a lot of the way, here. 00:25:26.170 --> 00:25:29.400 If they weren't orthogonal, then symmetric would have been dead, 00:25:29.400 --> 00:25:31.420 positive definite would have been dead, 00:25:31.420 --> 00:25:33.150 projection would have been dead. 00:25:33.150 --> 00:25:37.090 But those eigenvectors were orthogonal, 00:25:37.090 --> 00:25:40.180 so it came down to the eigenvalues. 00:25:40.180 --> 00:25:45.270 OK, that was like a chance to review a lot of this chapter. 00:25:50.790 --> 00:25:56.030 Shall I jump to the singular value decomposition, 00:25:56.030 --> 00:26:04.140 then, as the third, topic for, for the review? 00:26:04.140 --> 00:26:06.080 OK, so I'm going to. jump to this. 00:26:06.080 --> 00:26:06.580 OK. 00:26:13.950 --> 00:26:16.990 So this is the singular value decomposition, 00:26:16.990 --> 00:26:21.070 known to everybody as the SVD. 00:26:21.070 --> 00:26:27.720 And that's a factorization of A into orthogonal times 00:26:27.720 --> 00:26:33.830 diagonal times orthogonal. 00:26:33.830 --> 00:26:41.030 And we always call those U and sigma and V transpose. 00:26:41.030 --> 00:26:42.300 OK. 00:26:42.300 --> 00:26:46.660 And the key to that -- 00:26:46.660 --> 00:26:51.170 this is for every matrix, every A, every A. 00:26:51.170 --> 00:26:54.110 Rectangular, doesn't matter, whatever, 00:26:54.110 --> 00:26:56.740 has this decomposition. 00:26:56.740 --> 00:26:59.070 So it's really important. 00:26:59.070 --> 00:27:04.930 And the key to it is to look at things like A transpose A. 00:27:04.930 --> 00:27:07.360 Can we remember what happens with A transpose A? 00:27:07.360 --> 00:27:11.300 If I just transpose that I get V sigma transpose U 00:27:11.300 --> 00:27:15.420 transpose, that's multiplying A, which is U, 00:27:15.420 --> 00:27:24.850 sigma V transpose, and the result is V on the outside, 00:27:24.850 --> 00:27:27.990 s- U transpose U is the identity, 00:27:27.990 --> 00:27:30.930 because it's an orthogonal matrix. 00:27:30.930 --> 00:27:34.550 So I'm just left with sigma transpose sigma 00:27:34.550 --> 00:27:39.340 in the middle, that's a diagonal, possibly 00:27:39.340 --> 00:27:42.960 rectangular diagonal by its transpose, so the result, 00:27:42.960 --> 00:27:46.036 this is orthogonal, diagonal, orthogonal. 00:27:49.840 --> 00:27:55.620 So, I guess, actually, this is the SVD for A transpose A. 00:27:55.620 --> 00:27:59.930 Here I see orthogonal, diagonal, and orthogonal. 00:27:59.930 --> 00:28:00.440 Great. 00:28:00.440 --> 00:28:07.000 But a little more is happening. 00:28:07.000 --> 00:28:09.580 For A transpose A, the difference 00:28:09.580 --> 00:28:13.690 is, the orthogonal guys are the same. 00:28:13.690 --> 00:28:15.640 It's V and V transpose. 00:28:15.640 --> 00:28:17.290 What I seeing here? 00:28:17.290 --> 00:28:21.950 I'm seeing the factorization for a symmetric matrix. 00:28:21.950 --> 00:28:23.220 This thing is symmetric. 00:28:26.790 --> 00:28:30.590 So in a symmetric case, U is the same as V. 00:28:30.590 --> 00:28:33.210 U is the same as V for this symmetric matrix, 00:28:33.210 --> 00:28:34.990 and, of course, we see it happening. 00:28:34.990 --> 00:28:35.540 OK. 00:28:35.540 --> 00:28:39.760 So that tells us, right away, what V is. 00:28:39.760 --> 00:28:49.650 V is the eigenvector matrix for A transpose A. 00:28:49.650 --> 00:28:50.490 OK. 00:28:50.490 --> 00:28:57.070 Now, if you were here when I lectured about this topic, when 00:28:57.070 --> 00:29:00.600 I gave the topic on singular value decompositions, 00:29:00.600 --> 00:29:03.180 you'll remember that I got into trouble. 00:29:06.150 --> 00:29:09.860 I'm sorry to remember that myself, but it happened. 00:29:09.860 --> 00:29:10.550 OK. 00:29:10.550 --> 00:29:13.680 How did it happen? 00:29:13.680 --> 00:29:16.900 I was in great shape for a while, cruising along. 00:29:16.900 --> 00:29:20.120 So I found the eigenvectors for A transpose A. 00:29:20.120 --> 00:29:21.870 Good. 00:29:21.870 --> 00:29:24.800 I found the singular values, what were they? 00:29:24.800 --> 00:29:26.520 What were the singular values? 00:29:26.520 --> 00:29:32.720 The singular value number i, or -- 00:29:32.720 --> 00:29:36.890 these are the guys in sigma -- 00:29:36.890 --> 00:29:39.770 this is diagonal with the number sigma in it. 00:29:39.770 --> 00:29:42.910 This diagonal is sigma one, sigma two, 00:29:42.910 --> 00:29:46.090 up to the rank, sigma r, those are the non-zero ones. 00:29:48.830 --> 00:29:51.100 So I found those, and what are they? 00:29:51.100 --> 00:29:53.130 Remind me about that? 00:29:53.130 --> 00:29:59.630 Well, here, I'm seeing them squared, so their squares are 00:29:59.630 --> 00:30:03.470 the eigenvalues of A transpose A. 00:30:03.470 --> 00:30:04.760 Good. 00:30:04.760 --> 00:30:09.160 So I just take the square root, if I want the eigenvalues of A 00:30:09.160 --> 00:30:10.020 transpose -- 00:30:10.020 --> 00:30:11.990 If I want the sigmas and I know these, 00:30:11.990 --> 00:30:14.540 I take the square root, the positive square root. 00:30:14.540 --> 00:30:16.730 OK. 00:30:16.730 --> 00:30:20.610 Where did I run into trouble? 00:30:20.610 --> 00:30:25.220 Well, then, my final step was to find U. 00:30:25.220 --> 00:30:28.270 And I didn't read the book. 00:30:28.270 --> 00:30:35.480 So, I did something that was practically right, but -- 00:30:35.480 --> 00:30:38.880 well, I guess practically right is not quite the same. 00:30:38.880 --> 00:30:44.650 OK, so I thought, OK, I'll look at A A transpose. 00:30:44.650 --> 00:30:47.420 What happened when I looked at A A transpose? 00:30:47.420 --> 00:30:51.070 Let me just put it here, and then I can feel it. 00:30:51.070 --> 00:30:53.620 OK, so here's A A transpose. 00:30:57.120 --> 00:31:01.050 So that's U sigma V transpose, that's A, 00:31:01.050 --> 00:31:05.240 and then the transpose is V sigma transpose, 00:31:05.240 --> 00:31:06.300 U sigma transpose. 00:31:06.300 --> 00:31:07.930 Fine. 00:31:07.930 --> 00:31:10.610 And then, in the middle is the identity again, 00:31:10.610 --> 00:31:12.570 so it looks great. 00:31:12.570 --> 00:31:17.050 U sigma sigma transpose, U transpose. 00:31:17.050 --> 00:31:18.760 Fine. 00:31:18.760 --> 00:31:26.570 All good, and now these columns of U 00:31:26.570 --> 00:31:29.900 are the eigenvectors, that's U is the eigenvector 00:31:29.900 --> 00:31:33.120 matrix for this guy. 00:31:33.120 --> 00:31:36.960 That was correct, so I did that fine. 00:31:36.960 --> 00:31:38.600 Where did something go wrong? 00:31:38.600 --> 00:31:40.820 A sign went wrong. 00:31:40.820 --> 00:31:44.570 A sign went wrong because -- and now -- now I see, actually, 00:31:44.570 --> 00:31:49.140 somebody told me right after class, 00:31:49.140 --> 00:31:53.910 we can't tell from this description which sign to give 00:31:53.910 --> 00:31:55.200 the eigenvectors. 00:31:55.200 --> 00:32:00.570 If these are the eigenvectors of this matrix, 00:32:00.570 --> 00:32:02.660 well, if you give me an eigenvector 00:32:02.660 --> 00:32:04.790 and I change all its signs, we've 00:32:04.790 --> 00:32:06.920 still got another eigenvector. 00:32:06.920 --> 00:32:08.970 So what I wasn't able to determine 00:32:08.970 --> 00:32:13.940 (and I had a fifty-fifty change and life let me down,) 00:32:13.940 --> 00:32:16.600 the signs I just happened to pick 00:32:16.600 --> 00:32:19.070 for the eigenvectors, one of them 00:32:19.070 --> 00:32:21.750 I should have reversed the sign. 00:32:21.750 --> 00:32:27.190 So, from this, I can't tell whether the eigenvector 00:32:27.190 --> 00:32:31.150 or its negative is the right one to use in there. 00:32:31.150 --> 00:32:34.750 So the right way to do it is to, having 00:32:34.750 --> 00:32:38.640 settled on the signs, the Vs also, I 00:32:38.640 --> 00:32:42.100 don't know which sign to choose, but I choose one. 00:32:42.100 --> 00:32:43.220 I choose one. 00:32:43.220 --> 00:32:50.290 And then, instead, I should have used 00:32:50.290 --> 00:32:53.950 the one that tells me what sign to choose, the rule 00:32:53.950 --> 00:33:02.330 that A times a V is sigma times the U. 00:33:02.330 --> 00:33:07.140 So, having decided on the V, I multiply by A, 00:33:07.140 --> 00:33:09.640 I'll notice the factor sigma coming out, 00:33:09.640 --> 00:33:11.520 and there will be a unit vector there, 00:33:11.520 --> 00:33:17.310 and I now know exactly what it is, 00:33:17.310 --> 00:33:20.380 and not only up to a change of sign. 00:33:20.380 --> 00:33:22.390 So that's the good and, of course, 00:33:22.390 --> 00:33:25.910 this is the main point about the SVD. 00:33:25.910 --> 00:33:28.210 That's the point that we've diagonalized, 00:33:28.210 --> 00:33:32.950 that's A times the matrix of Vs equals 00:33:32.950 --> 00:33:37.710 U times the diagonal matrix of sigmas. 00:33:37.710 --> 00:33:39.470 That's the same as that. 00:33:39.470 --> 00:33:39.970 OK. 00:33:39.970 --> 00:33:47.800 So that's, like, correcting the wrong sign 00:33:47.800 --> 00:33:50.000 from that earlier lecture. 00:33:50.000 --> 00:33:52.810 And that would complete that, so that's how you would compute 00:33:52.810 --> 00:33:54.040 the SVD. 00:33:54.040 --> 00:33:58.380 Now, on the quiz, I going to ask -- well, maybe on the final. 00:33:58.380 --> 00:34:01.010 So we've got quiz and final ahead. 00:34:01.010 --> 00:34:05.400 Sometimes, you might be asked to find the SVD if I give you 00:34:05.400 --> 00:34:10.870 the matrix -- let me come back, now, to the main board -- 00:34:10.870 --> 00:34:17.880 or, I might give you the pieces. 00:34:17.880 --> 00:34:21.810 And I might ask you something about the matrix. 00:34:21.810 --> 00:34:31.580 For example, suppose I ask you, oh, let's say, 00:34:31.580 --> 00:34:36.590 if I tell you what sigma is -- 00:34:36.590 --> 00:34:37.460 OK. 00:34:37.460 --> 00:34:39.230 Let's take one example. 00:34:39.230 --> 00:34:43.820 Suppose sigma is -- 00:34:43.820 --> 00:34:46.350 so all that's how we would compute them. 00:34:46.350 --> 00:34:48.070 But now, suppose I give you these. 00:34:48.070 --> 00:34:52.320 Suppose I give you sigma is, say, three two. 00:34:57.130 --> 00:35:02.110 And I tell you that U has a couple of columns, 00:35:02.110 --> 00:35:04.580 and V has a couple of columns. 00:35:07.910 --> 00:35:10.330 OK. 00:35:10.330 --> 00:35:12.600 Those are orthogonal columns, of course, 00:35:12.600 --> 00:35:14.630 because U and V are orthogonal. 00:35:14.630 --> 00:35:16.120 I'm just sort of, like, getting you 00:35:16.120 --> 00:35:19.440 to think about the SVD, because we only had that one 00:35:19.440 --> 00:35:22.570 lecture about it, and one homework, 00:35:22.570 --> 00:35:28.460 and, what kind of a matrix have I got here? 00:35:28.460 --> 00:35:31.970 What do I know about this matrix? 00:35:31.970 --> 00:35:35.540 All I really know right now is that its singular values, 00:35:35.540 --> 00:35:39.390 those sigmas are three and two, and the only thing 00:35:39.390 --> 00:35:43.190 interesting that I can see in that is that they're not zero. 00:35:43.190 --> 00:35:48.390 I know that this matrix is non-singular, right? 00:35:48.390 --> 00:35:51.570 That's invertible, I don't have any zero eigenvalues, 00:35:51.570 --> 00:35:54.710 and zero singular values, that's invertible, 00:35:54.710 --> 00:36:02.290 there's a typical SVD for a nice two-by-two non-singular 00:36:02.290 --> 00:36:04.990 invertible good matrix. 00:36:04.990 --> 00:36:07.480 If I actually gave you a matrix, then you'd 00:36:07.480 --> 00:36:10.390 have to find the Us and the Vs as we just spoke. 00:36:10.390 --> 00:36:12.000 But, there. 00:36:12.000 --> 00:36:16.400 Now, what if the two wasn't a two but it was -- 00:36:16.400 --> 00:36:18.520 well, let me make an extreme case, here -- 00:36:18.520 --> 00:36:20.050 suppose it was minus five. 00:36:23.220 --> 00:36:24.810 That's wrong, right away. 00:36:24.810 --> 00:36:28.590 That's not a singular value decomposition, right? 00:36:28.590 --> 00:36:30.840 The singular values are not negative. 00:36:30.840 --> 00:36:36.011 So that's not a singular value decomposition, and forget it. 00:36:36.011 --> 00:36:36.510 OK. 00:36:36.510 --> 00:36:40.200 So let me ask you about that one. 00:36:40.200 --> 00:36:42.095 What can you tell me about that matrix? 00:36:45.090 --> 00:36:47.340 It's singular, right? 00:36:47.340 --> 00:36:50.480 It's got a singular matrix there in the middle, 00:36:50.480 --> 00:36:56.110 and, let's see, so, OK, it's singular, 00:36:56.110 --> 00:37:01.870 maybe you can tell me, its rank? 00:37:01.870 --> 00:37:04.320 What's the rank of A? 00:37:04.320 --> 00:37:08.900 It's clearly -- somebody just say it -- 00:37:08.900 --> 00:37:09.920 one, thanks. 00:37:09.920 --> 00:37:15.160 The rank is one, so the null space, 00:37:15.160 --> 00:37:18.850 what's the dimension of the null space? 00:37:18.850 --> 00:37:19.890 One. 00:37:19.890 --> 00:37:20.390 Right? 00:37:20.390 --> 00:37:23.940 We've got a two-by-two matrix of rank one, 00:37:23.940 --> 00:37:26.820 so of all that stuff from the beginning of the course 00:37:26.820 --> 00:37:30.310 is still with us. 00:37:30.310 --> 00:37:32.790 The dimensions of those fundamental spaces 00:37:32.790 --> 00:37:36.950 is still central, and a basis for them. 00:37:36.950 --> 00:37:40.890 Now, can you tell me a vector that's in the null space? 00:37:40.890 --> 00:37:47.000 And then that will be my last point to make about the SVD. 00:37:47.000 --> 00:37:49.400 Can you tell me a vector that's in the null space? 00:37:54.410 --> 00:38:00.820 So what would I multiply by and get zero, here? 00:38:00.820 --> 00:38:04.120 I think the answer is probably v2. 00:38:04.120 --> 00:38:07.820 I think probably v2 is in the null space, 00:38:07.820 --> 00:38:11.950 because I think that must be the eigenvector going 00:38:11.950 --> 00:38:14.800 with this zero eigenvalue. 00:38:14.800 --> 00:38:16.310 Yes. 00:38:16.310 --> 00:38:17.200 Have a look at that. 00:38:17.200 --> 00:38:21.390 And I could ask you the null space of A transpose. 00:38:21.390 --> 00:38:23.050 And I could ask you the column space. 00:38:23.050 --> 00:38:24.620 All that stuff. 00:38:24.620 --> 00:38:27.180 Everything is sitting there in the SVD. 00:38:27.180 --> 00:38:29.990 The SVD takes a little more time to compute, 00:38:29.990 --> 00:38:36.090 but it displays all the good stuff about a matrix. 00:38:36.090 --> 00:38:36.720 OK. 00:38:36.720 --> 00:38:39.680 Any question about the SVD? 00:38:39.680 --> 00:38:47.800 Let me keep going with further topics. 00:38:47.800 --> 00:38:48.990 Now, let's see. 00:38:48.990 --> 00:38:51.050 Similar matrices we've talked about, 00:38:51.050 --> 00:38:55.390 let me see if I've got another, -- 00:38:55.390 --> 00:38:57.620 OK. 00:38:57.620 --> 00:39:04.570 Here's a true false, so we can do that, easily. 00:39:04.570 --> 00:39:05.300 So. 00:39:05.300 --> 00:39:07.560 Question, A given. 00:39:10.480 --> 00:39:17.840 A is symmetric and orthogonal. 00:39:20.861 --> 00:39:21.360 OK. 00:39:26.920 --> 00:39:29.870 So beautiful matrices like that don't come along every day. 00:39:29.870 --> 00:39:36.480 But what can we say first about its eigenvalues? 00:39:36.480 --> 00:39:38.680 Actually, of course. 00:39:38.680 --> 00:39:41.810 Here are our two most important classes of matrices, 00:39:41.810 --> 00:39:45.500 and we're looking at the intersection. 00:39:45.500 --> 00:39:48.470 So those really are neat matrices, 00:39:48.470 --> 00:39:50.920 and what can you tell me about what could 00:39:50.920 --> 00:39:52.670 the possible eigenvalues be? 00:39:52.670 --> 00:39:57.540 Eigenvalues can be what? 00:39:57.540 --> 00:39:59.330 What do I know about the eigenvalues 00:39:59.330 --> 00:40:01.280 of a symmetric matrix? 00:40:01.280 --> 00:40:04.690 Lambda is real. 00:40:04.690 --> 00:40:06.550 What do I know about the eigenvalues 00:40:06.550 --> 00:40:10.260 of an orthogonal matrix? 00:40:10.260 --> 00:40:12.370 Ha. 00:40:12.370 --> 00:40:13.200 Maybe nothing. 00:40:13.200 --> 00:40:15.420 But, no, that can't be. 00:40:15.420 --> 00:40:17.911 What do I know about the eigenvalues of an orthogonal 00:40:17.911 --> 00:40:18.410 matrix? 00:40:18.410 --> 00:40:19.565 Well, what feels right? 00:40:22.330 --> 00:40:26.620 Basing mathematics on just a little gut instinct here, 00:40:26.620 --> 00:40:29.910 the eigenvalues of an orthogonal matrix 00:40:29.910 --> 00:40:33.800 ought to have magnitude one. 00:40:33.800 --> 00:40:36.280 Orthogonal matrices are like rotations, 00:40:36.280 --> 00:40:40.921 they're not changing the length, so orthogonal, the eigenvalues 00:40:40.921 --> 00:40:41.420 are one. 00:40:41.420 --> 00:40:50.070 Let me just show you why. 00:40:50.070 --> 00:40:53.920 So the matrix, can I call it Q for orthogonal 00:40:53.920 --> 00:40:55.810 Why? for the moment? 00:40:55.810 --> 00:40:58.760 If I look at Q x equal lambda x, how 00:40:58.760 --> 00:41:03.570 do I see that this thing has magnitude one? 00:41:03.570 --> 00:41:06.140 I take the length of both sides. 00:41:06.140 --> 00:41:08.930 This is taking lengths, taking lengths, 00:41:08.930 --> 00:41:13.520 this is whatever the magnitude is times the length of x. 00:41:13.520 --> 00:41:18.170 And what's the length of Q x if Q is an orthogonal matrix? 00:41:18.170 --> 00:41:20.910 This is something you should know. 00:41:20.910 --> 00:41:23.100 It's the same as the length of x. 00:41:23.100 --> 00:41:26.400 Orthogonal matrices don't change lengths. 00:41:26.400 --> 00:41:30.330 So lambda has to be one. 00:41:30.330 --> 00:41:31.090 Right. 00:41:31.090 --> 00:41:31.890 OK. 00:41:31.890 --> 00:41:34.440 That's worth committing to memory, 00:41:34.440 --> 00:41:36.930 that could show up again. 00:41:36.930 --> 00:41:37.430 OK. 00:41:37.430 --> 00:41:40.720 So what's the answer now to this question, 00:41:40.720 --> 00:41:42.940 what can the eigenvalues be? 00:41:42.940 --> 00:41:45.220 There's only two possibilities, and they 00:41:45.220 --> 00:41:55.050 are one and the other one, the other possibility 00:41:55.050 --> 00:42:00.770 is negative one, right, because these have the right magnitude, 00:42:00.770 --> 00:42:03.090 and they're real. 00:42:03.090 --> 00:42:04.070 OK. 00:42:04.070 --> 00:42:04.730 TK. 00:42:04.730 --> 00:42:07.090 true -- OK. 00:42:07.090 --> 00:42:08.260 True or false? 00:42:08.260 --> 00:42:12.220 A is sure to be positive definite. 00:42:12.220 --> 00:42:14.070 Well, this is a great matrix, but is it 00:42:14.070 --> 00:42:16.570 sure to be positive definite? 00:42:16.570 --> 00:42:17.090 No. 00:42:17.090 --> 00:42:19.380 If it could have an eigenvalue minus one, 00:42:19.380 --> 00:42:21.910 it wouldn't be positive definite. 00:42:21.910 --> 00:42:24.230 True or false, it has no repeated eigenvalues. 00:42:27.820 --> 00:42:29.700 That's false, too. 00:42:29.700 --> 00:42:31.710 In fact, it's going to have repeated eigenvalues 00:42:31.710 --> 00:42:33.740 if it's as big as three by three, 00:42:33.740 --> 00:42:35.630 one of these c- one of these, at least, 00:42:35.630 --> 00:42:37.270 will have to get repeated. 00:42:37.270 --> 00:42:37.770 Sure. 00:42:37.770 --> 00:42:40.040 So it's got repeated eigenvalues, but, 00:42:40.040 --> 00:42:42.910 is it diagonalizable? 00:42:42.910 --> 00:42:45.210 It's got these many, many, repeated eigenvalues. 00:42:45.210 --> 00:42:46.900 If it's fifty by fifty, it's certainly 00:42:46.900 --> 00:42:48.790 got a lot of repetitions. 00:42:48.790 --> 00:42:51.370 Is it diagonalizable? 00:42:51.370 --> 00:42:52.110 Yes. 00:42:52.110 --> 00:42:55.500 All symmetric matrices, all orthogonal matrices 00:42:55.500 --> 00:42:57.090 can be diagonalized. 00:42:57.090 --> 00:43:02.620 And, in fact, the eigenvectors can even be chosen orthogonal. 00:43:02.620 --> 00:43:05.120 So it could be, sort of, like, diagonalized 00:43:05.120 --> 00:43:09.270 the best way with a Q, and not just any old S. 00:43:09.270 --> 00:43:09.840 OK. 00:43:09.840 --> 00:43:11.820 Is it non-singular? 00:43:11.820 --> 00:43:15.275 Is a symmetric orthogonal matrix non-singular? 00:43:19.500 --> 00:43:21.790 Orthogonal matrices are always non-singular. 00:43:21.790 --> 00:43:22.900 Sure. 00:43:22.900 --> 00:43:26.910 And, obviously, we don't have any zero Eigenvalues. 00:43:26.910 --> 00:43:28.670 Is it sure to be diagonalizable? 00:43:28.670 --> 00:43:31.310 Yes. 00:43:31.310 --> 00:43:41.370 Now, here's a final step -- show that one-half of A plus I is A 00:43:41.370 --> 00:43:42.140 -- 00:43:42.140 --> 00:43:51.705 that is, prove one-half of A plus I is a projection matrix. 00:43:58.880 --> 00:43:59.380 OK? 00:44:04.080 --> 00:44:04.750 Let's see. 00:44:04.750 --> 00:44:05.510 What do I do? 00:44:09.170 --> 00:44:11.610 I could see two ways to do this. 00:44:11.610 --> 00:44:14.560 I could check the properties of a projection matrix, which 00:44:14.560 --> 00:44:15.700 are what? 00:44:15.700 --> 00:44:18.520 A projection matrix is symmetric. 00:44:18.520 --> 00:44:21.790 Well, that's certainly symmetric, because A is. 00:44:21.790 --> 00:44:24.440 And what's the other property? 00:44:24.440 --> 00:44:26.230 I should square it, and hopefully 00:44:26.230 --> 00:44:27.900 get the same thing back. 00:44:27.900 --> 00:44:32.230 So can I do that, square and see if I get the same thing back? 00:44:32.230 --> 00:44:37.060 So if I square it, I'll get one-quarter of A squared 00:44:37.060 --> 00:44:40.640 plus two A plus I, right? 00:44:40.640 --> 00:44:48.430 And the question is, does that agree with p- the thing itself? 00:44:48.430 --> 00:44:53.480 One-half A plus I. 00:44:53.480 --> 00:44:53.980 Hm. 00:44:57.240 --> 00:45:02.060 I guess I'd like to know something about A squared. 00:45:02.060 --> 00:45:03.260 What is A squared? 00:45:03.260 --> 00:45:05.090 That's our problem. 00:45:05.090 --> 00:45:06.260 What is A squared? 00:45:11.560 --> 00:45:13.510 If A is symmetric and orthogonal, 00:45:13.510 --> 00:45:17.390 A is symmetric and orthogonal. 00:45:22.060 --> 00:45:23.710 This is what we're given, right? 00:45:23.710 --> 00:45:27.990 It's symmetric, and it's orthogonal. 00:45:27.990 --> 00:45:31.350 So what's A squared? 00:45:31.350 --> 00:45:36.360 I. A squared is I, because A times A -- 00:45:36.360 --> 00:45:42.500 if A equals its own inverse, so A times A is the same as A 00:45:42.500 --> 00:45:46.150 times A inverse, which is I. 00:45:46.150 --> 00:45:52.320 So this A squared here is I. 00:45:52.320 --> 00:45:54.530 And now we've got it. 00:45:54.530 --> 00:45:57.810 We've got two identities over four, that's good, 00:45:57.810 --> 00:46:01.060 and we've got two As over four, that's good. 00:46:01.060 --> 00:46:01.670 OK. 00:46:01.670 --> 00:46:05.960 So it turned out to be a projection matrix safely. 00:46:05.960 --> 00:46:08.310 And we could also have said, well, 00:46:08.310 --> 00:46:11.230 what are the eigenvalues of this thing? 00:46:11.230 --> 00:46:14.870 What are the eigenvalues of a half A plus I? 00:46:14.870 --> 00:46:17.970 If the eigenvalues of A are one and minus one, 00:46:17.970 --> 00:46:21.840 what are the eigenvalues of A plus I? 00:46:21.840 --> 00:46:25.860 Just stay with it these last thirty seconds here. 00:46:25.860 --> 00:46:29.140 What if I know these eigenvalues of A, 00:46:29.140 --> 00:46:31.480 and I add the identity, the eigenvalues 00:46:31.480 --> 00:46:35.600 of A plus I are zero and two. 00:46:35.600 --> 00:46:39.480 And then when I divide by two, the eigenvalues are zero and 00:46:39.480 --> 00:46:40.020 one. 00:46:40.020 --> 00:46:43.130 So it's symmetric, it's got the right eigenvalues, 00:46:43.130 --> 00:46:45.070 it's a projection matrix. 00:46:45.070 --> 00:46:49.290 OK, you're seeing a lot of stuff about eigenvalues, 00:46:49.290 --> 00:46:54.460 and special matrices, and that's what the quiz is about. 00:46:54.460 --> 00:46:57.230 OK, so good luck on the quiz.